The Forgotten History of Wave Function Variance Optimization and its Relevance for Neural-Network VMC

• URL: http://arxiv.org/abs/2609.08453v1
• Date: Tue, 08 Sep 2026 08:53:17 GMT
• Title: The Forgotten History of Wave Function Variance Optimization and its Relevance for Neural-Network VMC
• Abstract summary: Minimizing variance implicitly assumes a Gaussian local energy distribution.<n>For Coulombic systems, the local energy distribution exhibits $E-4$ power-law tails, causing the Central Limit Theorem to fail for the variance estimator.<n>We benchmark these functionals on $H+$, an exactly solvable system at every internuclear distance.<n>While energy minimization by construction yields the lowest energy, variance minimization is surpassed at every $R$ by alternative functionals.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We revisit the almost century-old question of which functional of the local energy best optimizes a trial wave function, a problem of central importance in Variational Monte Carlo (VMC) and, more recently, in Neural-Network VMC (NN-VMC). While variance optimization dates back to the 1930s, the high statistical noise and heavy-tailed local energy distributions inherent to modern neural-network wave functions have renewed interest in this approach. We retrace its long and largely forgotten history here, showing its direct relevance to modern Neural Quantum States (NQS) frameworks. Minimizing the variance (an $L^2$ norm) implicitly assumes a Gaussian local energy distribution: an unjustified assumption. For Coulombic systems, the local energy distribution exhibits $E^{-4}$ power-law tails, causing the Central Limit Theorem to fail for the variance estimator. This instability can be mitigated by robust cost functions: the Mean Absolute Deviation (MAD, an $L^1$ norm), the Cauchy loss, or the $L_{-4}$ functional, which features a tail analytically designed to match the $E^{-4}$ exponent. We benchmark these functionals on $H_2^+$, an exactly solvable system at every internuclear distance, using the Guillemin-Zener wave function across the full potential energy curve. While energy minimization by construction yields the lowest energy, variance minimization is surpassed at every $R$ by alternative functionals: MAD proves superior in the bonding region, while $L_{-4}$ performs best in the dissociation regime.

Related papers

• Maximum Entropy Encoding of Energy-Weighted Spherical Moments [2.1503190497152453]
  We study how angular energy signals composed of non-negative Monte Carlo path samples can be compressed and reconstructed for irradiance using finite moments.<n>Experiments cover 981 Poly Haven HDRI 2K scenes and three Debevec probes.
  arXiv  Detail & Related papers  (2026-08-20T00:58:59Z)
• Is Variational Monte Carlo Robust? Sharp Moment Thresholds and Heavy-tailed Stochastic Optimization [3.4519796338615225]
  Variational Carlo (VMC) is a central algorithm in electronic structure theory.<n>In this work, we show that the resulting optimization problem is intrinsically governed by the nodal geometry of the wave function.<n>We introduce a new robust variant of VMC based on clipping both the local energy and a random variable.
  arXiv  Detail & Related papers  (2026-06-24T16:29:54Z)
• Derivative Informed Learning of Exchange-Correlation Functionals [46.44973075720396]
  Machine-learned (ML) exchange-correlation (XC) functionals aim to replace human-designed density functional approximations.<n>We study a hybrid-distillation setting in which $mathcalO(N4)$-scaling ML-XC functionals are trained to reproduce B3LYP/def2-SVP targets.<n>We introduce Derivative Informed XC-Loss (DI-Loss), a loss that incorporates additional information from the reference hybrid functional.
  arXiv  Detail & Related papers  (2026-06-02T23:07:54Z)
• Wiener Chaos Expansion based Neural Operator for Singular Stochastic Partial Differential Equations [58.188204772101756]
  This paper explores how our recently developed Wiener Chaos Expansion (WCE)-based neural operator (NO) can be applied to singular partial differential equations.<n>We leverage feature-wise linear modulation (FiLM) to appropriately capture the dependency between the solution of singular SPDE and its smooth remainder.<n>The resulting WCE-FiLM-NO shows excellent performance on $boldsymbol4$, as measured by relative $L$ loss, out-of-distribution and autocorrelation score.
  arXiv  Detail & Related papers  (2026-03-09T10:50:30Z)
• Proving the Limited Scalability of Centralized Distributed Optimization via a New Lower Bound Construction [57.93371273485736]
  We consider a centralized distributed learning setup where all workers jointly find an unbiased bound LDeltaepsilon2,$ better poly-logarithmically in $n$, even in the homogeneous (i.i.d.) case, where all workers access the same distribution.
  arXiv  Detail & Related papers  (2025-06-30T13:27:39Z)
• Optimal policy evaluation using kernel-based temporal difference methods [78.83926562536791]
  We use kernel Hilbert spaces for estimating the value function of an infinite-horizon discounted Markov reward process. We derive a non-asymptotic upper bound on the error with explicit dependence on the eigenvalues of the associated kernel operator. We prove minimax lower bounds over sub-classes of MRPs.
  arXiv  Detail & Related papers  (2021-09-24T14:48:20Z)
• Universality class of Ising critical states with long-range losses [0.0]
  We show that spatial resolved dissipation can act on $d$-dimensional spin systems in the Ising universality class. We consider power-law decaying spin losses with a Lindbladian spectrum closing at small momenta as $propto qalpha$.
  arXiv  Detail & Related papers  (2021-08-27T17:59:51Z)
• Dynamics of Open Quantum Systems II, Markovian Approximation [0.0]
  We show that for fixed, small values of the coupling constant $lambda$, the true reduced dynamics of the system is approximated by the Davies-Lindblad generator. The difference between the true and the Markovian dynamics is $O(lambda|1/4)$ for all times.
  arXiv  Detail & Related papers  (2021-04-30T18:09:35Z)
• Finding Global Minima via Kernel Approximations [90.42048080064849]
  We consider the global minimization of smooth functions based solely on function evaluations. In this paper, we consider an approach that jointly models the function to approximate and finds a global minimum.
  arXiv  Detail & Related papers  (2020-12-22T12:59:30Z)
• Variational Monte Carlo calculations of $\mathbf{A\leq 4}$ nuclei with an artificial neural-network correlator ansatz [62.997667081978825]
  We introduce a neural-network quantum state ansatz to model the ground-state wave function of light nuclei. We compute the binding energies and point-nucleon densities of $Aleq 4$ nuclei as emerging from a leading-order pionless effective field theory Hamiltonian.
  arXiv  Detail & Related papers  (2020-07-28T14:52:28Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.